Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Sign up to join this community
From reading answers here, I've noticed that some people write integrals as $\int dx \; f(x)$, while other people write them as $\int f(x)\;dx$.
I realize that there is no mathematical difference between the two notation forms, but was wondering why some people choose the first method over the second. Is there some place in higher maths that it becomes beneficial to write the differential first?
(I, personally, have always used the second method, just because I was taught that way...)
When you have a lot of integrals, particularly with limits, it can be very helpful at times to be able to tell at a glance which integral is over which variable.
$$\int_0^1 \int_2^3 f(x,y) \; \mathrm d x \mathrm d y$$
This is not particularly readable or clear, especially when $f$ is lengthy and there are more nested integrals etc. I could also imagine it being misinterpreted.
By contrast,
$$\int_0^1\mathrm d y \int_2^3 \mathrm d x \; f(x,y)$$
makes it very clear what is going on. The only price you pay is possible ambiguity about where the integral ends, but this is easier to make clear with formatting and less of an issue anyway.
Edit: It also just occurred to me that the second notation ties in better with the syntax of an operator. That is, if one thinks of $\int_0^1 \mathrm d x$ as being an operator, taking a function to its integral, it's more natural to have the whole operator together in one lump. Think of how one changes $$\frac {\partial f}{\partial x}\to \frac{\partial}{\partial x} f$$
In my opinion, the symbol $\int \mathrm{d}x\, f(x)$ is simply bad notation. On the contrary, when dealing with (at least) double integrals, the advantage of $$\int_{a}^b \mathrm{d}x \int_{c}^{d} \mathrm{d}y\, f(x,y)$$ is that it perfectly fits into the stategy of Fubini's theorem. Imagine that you are computing a double integral: you say "Ok, now I fix the $x$ variable and integrate with respect to the $y$ variable." It is natural to write down $x$ first.
This said, I think that $$\int_{a}^{b} \left( \int_{c}^{d} f(x,y)\, \mathrm{d}y \right) \mathrm{d}x$$ should be used in printed papers and books.
Writing the thing with a differential first its useful in some contexts, by example in measure theory when $K:\mathcal{X}\times \mathcal{F} \to \mathbb{R}$ is a transition kernel, that is when $(\Omega ,\mathcal{F})$ is a measurable space and the map $A\mapsto K(x,A)$ is a measure for every $x\in \mathcal{X}$ and $x\mapsto K(x,A)$ is measurable for every $A\in \mathcal{F}$, then its common to define $$ Kf(x):=\int f(\omega )K(x,d\omega) $$ where above the "differential" notation says what is the measure of integration. Thus its more useful to write the integrals with the measure before the function to have the same order as in the LHS, so it becomes a useful mnemothecnic rule to write $$ Kf(x)=\int K(x,d\omega)f(\omega ) $$ Similarly, if $\mu$ is a measure we can define the measure $$ \mu K(A):=\int K(x,A)\mu(dx) $$ where again its more useful to write the measure before the kernel in the integral, that is $\int \mu(dx)K(x,A)$ instead. Thus from here its easy to remember what means things like $MK$, where $M$ is another transition kernel, or $\mu Kf$, that is $$ MK(x,A)=\int M(x,d\omega)K(\omega ,A),\quad \mu K f=\int \mu(dx)K(x,d\omega)f(\omega ) $$ I saw this use in the book on probability theory of Erhan Çinlar.
Also a justification for the use in more classical contexts, as a Riemann integral, is that the integration of continuous functions is a linear map, such that we can define $$ Tf:=\int_{a}^b f(x) dx $$ But then in some sense it seems more cleaner to write $$ Tf=\int_{a}^b dx\, f(x) $$ so in some sense $\int_{a}^b dx\,\cdot\, $ represents the linear map $T$.
Apart from the idea that $\int_a^b$ should be next to the $\mathrm{d}x$ to make clear we mean $\int_{x=a}^{x=b}$, or the fact that $\int_a^b\mathrm{d}x$ is a linear operator that deserves a symbol, there are some ways such placement protects us from our own mistakes:
One advantage of writing $\mathrm{d}x$ first is you won't forget it, which we mustn't do if we have to juggle multiple variables, e.g. due to a substitution. Sometimes, you have to include quite a few terms or factors in the integrand.
If you're handwriting an integral (which may sound antiquated, but every lecture attendee knows it isn't), writing $\mathrm{d}x$ second carries the further risk of running out of room. You could avoid that by not writing the $\mathrm{d}x$ until you're done, but then you might forget it.
